Given q1, q2 ∈ C \ {0}, we construct a unital Banach algebra Bq1,q2 which contains a universal normalized solution to the (q1, q2)-deformed Heisenberg–Lie commutation relations in the following specific sense: (i) Bq1,q2 contains elements b1, b2, and b3 which satisfy the (q1, q2)-deformed Heisenberg–Lie commutation relations (that is, b1b2 − q1b2b1 = b3, q2b1b3 − b3b1 = 0, and b2b3 − q2b3b2 = 0...

Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate (n + 1)-form. The case n = 2 is relevant to string theory: we call this ‘2-plectic geometry.’ Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the ...

A Lie bialgebra is a vector space endowed simultaneously with the structure of algebra and coalgebra, some compatibility condition. Moreover, brackets have skew symmetry. Because close relation between bialgebras quantum groups, it interesting to consider structures on L related Virasoro algebra. In this paper, are investigated by computing Der(L, L⊗L). It proved that all such triangular coboun...

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...

Given a complex projective algebraic variety, write H•(X,C) for its cohomology with complex coefficients and IH •(X,C) for its Intersection cohomology. We first show that under some fairly general conditions the canonical map H•(X,C) → IH •(X,C) is injective. Now let Gr := G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. ...

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley–Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that M1,M2 ∈ L =⇒ M1M2 − M2M1 ∈ L. Two matrix Lie algebras are conjugate if there is an invertible matrix M such that L1 = ML...

In this appendix we give another algebraic proof of the Weyl character formula, using methods that have many other applications in Lie theory. We begin by setting up the machinery of Lie algebra cohomology (without assuming any previous background in homological algebra). We define the cohomology spaces for a Lie algebra representation in terms of a cochain complex and differential, with the co...

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

Given a complex projective algebraic variety, write H•(X,C) for its cohomology with complex coefficients and IH •(X,C) for its Intersection cohomology. We first show that under some fairly general conditions the canonical map H•(X,C) → IH •(X,C) is injective. Now let Gr := G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. ...